Question

if ATA-1 is symmetric then P.T=(AT)2​

Answer 1

SOLUTION

TO DETERMINE

$\sf{If \: \: {A}^{T} {A}^{ - 1} \: is \: symmetric \: \: then \: \: { A}^{2} = }$

EVALUATION

Here it is given that

$\sf{ {A}^{T} {A}^{ - 1} \: is \: symmetric \: \: }$

$\sf{ \implies \: {({A}^{T} {A}^{ - 1})}^{T} = {A}^{T} {A}^{ - 1} }$

$\sf{ \implies \: {({A}^{ - 1})}^{T} {({A}^{T} )}^{T}= {A}^{T} {A}^{ - 1} }$

$\sf{ \implies \: {({A}^{ - 1})}^{T} A= {A}^{T} {A}^{ - 1} \: \: \bigg[\because \: \: {({A}^{T} )}^{T} = A\bigg]}$

$\sf{ \implies \: {({A}^{ - 1})}^{T} AA= {A}^{T} {A}^{ - 1} A\: \: }$

$\sf{ \implies \: {({A}^{ - 1})}^{T} AA= {A}^{T}I \: \: \bigg[\because \: \: {{A}^{ - 1} } A= I \: \bigg]}$

$\sf{ \implies \: {({A}^{ - 1})}^{T} AA= {A}^{T}}$

$\sf{ \implies \: {({A}^{ - 1})}^{T} {A }^{2} = {A}^{T}}$

$\sf{ \implies \: {A}^{T}{({A}^{ - 1})}^{T} {A }^{2} ={A}^{T} {A}^{T}}$

$\sf{ \implies \: {A}^{T}{({A}^{ T})}^{ - 1} {A }^{2} ={A}^{T} {A}^{T}} \: \bigg[ \because \: {({A}^{ T})}^{ - 1} = {({A}^{ - 1})}^{T} \bigg]$

$\sf{ \implies \: I.{A }^{2} ={({A}^{T})}^{2} \: \: \: \: \bigg[ \: \because \:{A}^{T}{({A}^{ T})}^{ - 1} = I\bigg]}$

$\sf{ \implies \: {A }^{2} ={({A}^{T})}^{2} }$

FINAL ANSWER

$\boxed{ \: \: \: \sf{ \: {A }^{2} ={({A}^{T})}^{2} } \: \: \: }$

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